How do you prove the fundamental theorem of calculus

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The Fundamental Theorem of Calculus establishes the profound connection between differentiation and integration. It consists of two parts. Part 1 states that if f is continuous on an interval, and we define a function F as the integral of f from a fixed point a to a variable point x, then F is differentiable and its derivative equals the original function f. Part 2 states that if f is continuous and G is any antiderivative of f, then the definite integral of f from a to b equals G of b minus G of a. This theorem is remarkable because it shows that differentiation and integration are inverse operations. Let's prove Part 1 of the Fundamental Theorem of Calculus. We start with the definition of the derivative of F(x) as the limit of the difference quotient as h approaches zero. Substituting the definition of F, we get the difference of two integrals, which simplifies to the integral of f from x to x+h. By the Mean Value Theorem for integrals, this equals f(c) times h, where c is some point between x and x+h. In the difference quotient, the h's cancel out, leaving us with f(c). As h approaches zero, c approaches x, and by the continuity of f, the limit equals f(x). Therefore, F'(x) equals f(x), which completes our proof of Part 1. Now let's prove Part 2 of the Fundamental Theorem of Calculus. We start by defining F(x) as the integral of f from a to x. From Part 1, we know that F'(x) equals f(x). We're given that G is any antiderivative of f, so G'(x) also equals f(x). Since F and G have the same derivative, they must differ by a constant: G(x) equals F(x) plus C. Substituting the definition of F, we get G(x) equals the integral of f from a to x, plus C. To find C, we evaluate at x equals a. The integral from a to a is zero, so G(a) equals C. Substituting back, we get G(x) equals the integral of f from a to x, plus G(a). Evaluating at x equals b, we get G(b) equals the integral of f from a to b, plus G(a). Rearranging, we get the integral of f from a to b equals G(b) minus G(a), which completes our proof of Part 2. The Fundamental Theorem of Calculus has numerous important applications. First, it provides a practical method for computing definite integrals: find an antiderivative F of the function f, then calculate F(b) minus F(a). Second, it gives us a way to calculate the area under a curve, which is represented by the definite integral. Third, it introduces the concept of accumulation functions, where F(x) equals the integral of f from a to x, representing how a quantity accumulates over time or space. In physics, the theorem establishes the relationship between position and velocity: velocity is the derivative of position, and position is the integral of velocity. Similarly, work is the integral of force with respect to distance. These applications demonstrate why the Fundamental Theorem is considered one of the most important results in calculus, connecting the two major branches of the subject. To summarize what we've learned about the Fundamental Theorem of Calculus: First, it establishes the profound connection between the two major branches of calculus - differentiation and integration. Part 1 states that if we define F(x) as the integral of f from a to x, then the derivative of F equals the original function f. Part 2 gives us a practical method for computing definite integrals by finding an antiderivative G of f and evaluating G(b) minus G(a). This theorem is revolutionary because it proves that differentiation and integration are inverse operations. It also provides the mathematical foundation for numerous applications in physics, engineering, and other fields where we need to calculate accumulations or rates of change.